draft unit plan 7.rp.1-3: analyze proportional...

DRAFT UNIT PLAN7.RP.1-3: Analyze Proportional Relationships and Use Them to Solve Real-World and Mathematical Problems

Overview: The overview statement is intended to provide a summary of major themes in this unit.

This unit extends knowledge of fractions and ratios from previous grades, and it develops understanding of proportionality to solve single-step and multi-step problems, and to distinguish proportional relationships from other relationships (see Key Advances from Previous Grades). Knowledge and understanding of ratios and proportionality can be used to solve a variety of problems, including percent, scale drawings, and unit rate. Graphs of proportional relationships extend unit rate and connect it to slope of a line.

Teacher Notes: The information in this component provides additional insights which will help educators in the planning process for this unit.

As explained in the draft Progressions for the Common Core State Standards in Mathematics (10 September 2011), two instructional perspectives provide insight for the instruction of ratios in relation to proportional reasoning: (1) ratio as a composed unit or batch, and (2) ratio as a fixed numbers of parts.

Ratio is a unit or batch, for example, there are 3 cups of apple juice for every 2 cups of grape juice in the mixture. This way uses a composed unit: 3 cups apple juice and 2 cups grape juice. Any mixture that is made from some number of the composed unit is in the ratio 3 to 2. In the table, each of the mixtures of apple juice and grape juice are combined in a ratio of 3 to 2:

Ratio is a combined number of parts, for example a mixture is made from 3 parts

apple juice and 2 parts rape juice, where all parts are the same size, but can be any unit.

apple juice:

grape juice:

Draft Maryland Common Core State Curriculum Lesson Plan for Grade 7 Mathematics December 2011 of 38

Each part represents the same amount, but can be any amount, such as 2 cups or 5 liters

# cups apple juice 3 6 9 12

32 1

#cups grape juice2 4 6 8 1

23

2 composed units 12

of a composed unit


If 1 part is: 1 cup 2 cups 5 liters 3 quarts Amount of apple juice: 3 cups 6 cups 15 liters 9 quarts In the table, each mixture of apple juice to grape juice is in a Amount of grape juice: 2 cups 4 cups 10 liters 6 quarts proportional relationship of 3 to 2, regardless of the unit.

Enduring Understandings: Enduring understandings go beyond discrete facts or skills. They focus on larger concepts, principles, or processes. They are transferable and apply to new situations within or beyond the subject.

At the completion of the unit on Ratios and Proportional Reasoning, the student will understand that: A ratio is a multiplicative comparison of two quantities. Proportional reasoning involves relationships among ratios. Proportional situations involve multiplicative relationships. Situations based on proportional reasoning can involve multiple representations. Ratios and proportional reasoning interconnects with topics in geometry, probability, and rational numbers.

Essential Question(s): A question is essential when it stimulates multi-layered inquiry, provokes deep thought and lively discussion, requires students to consider alternatives and justify their reasoning, encourages re-thinking of big ideas, makes meaningful connections with prior learning, and provides students with opportunities to apply problem-solving skills to authentic situations.

What are the types/varieties of situations in life that depend on or require the application of ratios and proportional reasoning?

Priority Clusters: According to the Partnership for the Assessment of Readiness for College and Careers (PARCC), some clusters require greater emphasis than others. The table below shows PARCC’s relative emphasis for each cluster. Prioritization does not imply neglect or exclusion of material. Clear priorities are intended to ensure that the relative importance of content is properly attended to. Note that the prioritization is in terms of cluster headings.

Key: ■ Major Clusters Supporting Clusters Additional Clusters

Ratios and Proportional Reasoning■Analyze proportional relationships and use them to solve real-world and mathematical problems.

The Number System■ Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.



Expressions and Equations■ Use properties of operations to generate equivalent expressions.

■ Solve real-life and mathematical problems using numerical and algebraic expressions and equations.Geometry Draw, construct, and describe geometrical figures and describe the relationships between them. Solve real-life and mathematical problems involving angle measure, area surface area, and volume.

Statistics and Probability Use random sampling to draw inferences about a population.Draw informal comparative inferences about two populations. Investigate chance processes and develop, use, and evaluate probability models.

Focus Standards (Listed as Examples of Opportunities for In-Depth Focus in the PARCC Content Framework document): According to the Partnership for the Assessment of Readiness for College and Careers (PARCC), this component highlights some individual standards that play an important role in the content of this unit. Educators may choose to give the indicated mathematics an especially in-depth treatment, as measured for example by the number of days; the quality of classroom activities for exploration and reasoning; the amount of student practice; and the rigor of expectations for depth of understanding or mastery of skills.

7.RP.2 Students in grade 7 need ample opportunity to refine their ability to recognize, represent, and analyze proportional relationships in various ways, including through the use of tables, graphs, and equations.

7.EE.3 Applying properties of operations to calculate with rational numbers in any form while using tools strategically, converting between forms as appropriate, and assessing the reasonableness of answers using mental computation and estimation strategies is a major capstone standard for arithmetic and its applications.

Possible Student Outcomes: The following list is meant to provide a number of achievable outcomes that apply to the lessons in this unit. The list does not include all possible student outcomes for this unit, nor is it intended to suggest sequence or timing. These outcomes should depict the content segments into which a teacher might elect to break a given standard. They may represent groups of standards that can be taught together.

The student will: Determine, represent, and calculate proportional relationships between quantities in tables, graphs, equations, diagrams, and verbal

descriptions. Compute unit rates and distinguish them from other ratios.



Use proportional relationships to solve problems involving simple interest, tax, markups and markdowns, gratuities, commissions, fees, percent increase and decrease, percent error, scale drawings, and probability, among others.

Solve problems involving unit rates associated with ratios of fractions (e.g., if a person walks ½ mile in each ¼ hour, the unit rate is the complex fraction ½ / ¼ miles per hour or 2 miles per hour)

Analyze proportional relationships in geometric figures. Approximate the probability of simple and complex events.

Progressions from Common Core State Standards in Mathematics: For an in-depth discussion of the overarching, “big picture” perspective on student learning of content related to this unit, see:

The Common Core Standards Writing Team (10 September 2011). Progressions for the Common Core State Standards in Mathematics (draft), accessed at: http://commoncoretools.files.wordpress.com/2011/09/ccss_progression_rp_67_2011_11_121.pdf

Vertical Alignment: Vertical curriculum alignment provides two pieces of information: (1) a description of prior learning that should support the learning of the concepts in this unit, and (2) a description of how the concepts studied in this unit will support the learning of additional mathematics.

Key Advances from Previous Grades: Students enlarge their concept of and capability with ratios and proportional reasoning by:o understanding of and skill with multiplication, division, decimals, fractions regarding the study of ratios

Additional Mathematics: Students will use ratios and proportional reasoning skills: o in grade 8 when graphing proportional relationships and interpreting unit rate as the slope of a line (constant rate of change, direct

variation)o in geometry and in algebra when studying similar figures and slopes of lineso in trigonometry when studying sine, cosine, tangent, and other trigonometric ratioso in algebra for situations involving constant rates of changeo in calculus when working with average and instantaneous rates of change of functions o in statistics when describing statistical data

Possible Organization of Unit Standards: This table identifies additional grade-level standards within a given cluster that further define and support the overarching unit standards from within the same cluster. The table also provides instructional connections to grade-level standards from outside the cluster.


http://commoncoretools.files.wordpress.com/2011/09/ccss_progression_rp_67_2011_11_121.pdf


Overarching Unit Standards

Supporting Standards within the Cluster

Instructional Connections outside the Cluster

7.RP.1: Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like and/or different units.

7.G.1: Solve problems involving scale drawings of

geometric figures Including computing actual lengths and areas from

a scale drawing Reproduce a scale drawing at a different scale.

7.SP.6: Approximate the probability of a chance event by

collecting data on the process that produces it Observe its long-run relative frequency Predict the approximate relative frequency given

the probability.7.SP.8: Find probabilities of compound events Use organized lists, tables, tree diagrams, and

simulation.7.RP.2: Recognize and represent proportional relationships between quantities.

7.RP.2a: Decide whether two quantities are in a

proportional relationship by testing for equivalent ratios in a table or graphing on a coordinate plane

Observe whether the graph is a straight line through the origin.

7.RP.2b: Identify the constant of proportionality

(unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

7.RP.2c: Represent proportional relationships by

equations.

7.RP.2d:



Overarching Unit Standards

Supporting Standards within the Cluster

Instructional Connections outside the Cluster

Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.

7.RP.3: Use proportional relationships to solve multi-step ratio and percent problems.

7.EE.3: Solve multi-step real-life and mathematical

problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically;

Apply properties of operations to calculate with numbers in any form

Convert between forms as appropriate Assess the reasonableness of answers using

mental computation and estimation strategies.

Connections to the Standards for Mathematical Practice: This section provides samples of how the learning experiences for this unit support the development of the proficiencies described in the Standards for Mathematical Practice. The statements provided offer a few examples of connections between the Standards for Mathematical Practice in the content standards of this unit. The list is not exhaustive and will hopefully prompt further reflection and discussion.

In this unit, educators should consider implementing learning experiences which provide opportunities for students to:

1. Make sense of problems and persevere in solving them. Recognize that multiplicative relationships are proportional, including situations found in similar polygons and probability. Decide whether two quantities are in a proportional or additive relationship. Decide how to correctly set up a proportion in various formats. Decide if a proportional relationship in a problem is best represented by tables, graphs, equations, diagrams, or verbal descriptions.

2. Reason abstractly and quantitatively Compute unit rates with complex fractions.



3. Construct Viable Arguments and critique the reasoning of others. Recognize and justify the difference between a unit a rate and a ratio. Recognize and justify that two equal ratios represent a proportion.

4. Model with Mathematics

Proportional relationships present opportunities for modeling. For example, the number of people who live in an apartment building might be taken as proportional to the number of stories in the building for modeling purposes.

Construct a nonverbal representation of a verbal problem. Construct a graph to determine the proportional relationship of a situation. Draw a diagram to represent a proportional relationship. For example, draw similar polygons that are in proportion to one another.

5. Use appropriate tools strategically Use technology or manipulatives to explore a problem numerically or graphically.

6. Attend to precision Use mathematics vocabulary (i.e., ratio, unit rate, complex fraction, proportional relationships) properly when discussing problems. Demonstrate their understanding of the mathematical processes required to solve a problem by carefully showing all of the steps in

solving the problem. Label final answers appropriately. Label graphs and tables.

7. Look for and make use of structure. Make observations about how ratios and proportional relationships are set up to decide which configurations (e.g., “between ratios”

versus “within ratios”) are possible ways to solve a problem.

8. Look for and express regularity in reasoning Pay special attention to (0,0) and (1,r) when graphing a proportional relationship.

Content Standards with Essential Skills and Knowledge Statements and Clarifications: The Content Standards and Essential Skills and Knowledge statements shown in this section come directly from the Maryland State Common Core Curriculum Frameworks. Clarifications were added as needed. Please note that only the standards or portions of standards that needed further explanation have supporting statements. Educators should be cautioned against perceiving this as a checklist. All information added is intended to help the reader gain a better understanding of the standards.

Standard Essential Skills and Knowledge Clarification


Within Ratios


7.RP.1: Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like and or different units. (i.e., if a

person walks 12

mile in each

14

hour, compute the unit rate

as the complex fraction

1214

miles per hour or 2 miles per hour).

7.RP.2: Recognize and represent proportional relationships between quantities.

.

Ability to describe and identify complex fractions (see 7.NS.3)

Ability to recognize the difference(s) between a unit rate and a ratio (see 7.G.1)

Ability to recognize in a given proportional situation that the two “within ratios” and the two “between ratios” are the same

Within Ratios: AB

= ab

or BA

= ba

Ratio as a complex fraction:

1214

milehour

Ratio as a fraction: 42

mileshours

Ratio as a unit rate: 21

mileshour

All three of these ratios are equivalent; thus, in proportion to one another.

7.RP.2a: “within” ratios vs. “between” ratios If a person walks 4 miles in 2 hours, how many miles will she walk in 3 hours? Let m = miles

Set up of the proportion using “within ratios” format:

42

= m3

or 24

= 3m

Set up of the proportion using “between ratios” format:


A

B


Between Ratios: Aa

= Bb

or aA

= bB

4m

= 23

or m4

= 32

2a: Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin

Ability to distinguish between additive and multiplicative situations

Ability to recognize that two equal ratios represent a proportion

Ability to recognize and represent the connection between equivalent ratios, values in a table, and graphed ordered pairs (see 7.G.1)

7.RP.2a: additive reasoning versus multiplicative reasoning. Proportional situations are based on multiplicative relationships. Equal ratios result from multiplication or division, not addition or subtraction.

Consider any proportion, for example 17

= 535

. What operation was used to

convert 17

to 5

35? Was additive reasoning used:

17

+ 55

= 4035

? Or was

multiplicative reasoning used: 17 •

55=

535?

The example below shows multiplicative reasoning:

Last month, two bean plants were measured at 9 inches tall and 15 inches tall. Today they are 12 inches tall and 18 inches tall, respectively. Which bean plant grew more during the month, the 9-inch bean plant or the 15-inch bean plant?

Using additive reasoning, we know that each plant added 3 inches in a month.



To determine which plant grew more, however, we use multiplicative reasoning. In other words, we want to know what proportion of the original plant height is represented by 3 inches.

The 3-inch increase represents 39

or 13

of the first plant’s growth, whereas the

3-inch increase represents 315

or15

of the second plant’s growth. Since 39

> 3

15

(or13 >

15¿, the 9-inch plant grew more.

2b: Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams and verbal descriptions of proportional relationships.

Ability to express unit rates using a variety of representations, given a contextual situation

2c: Represent proportional relationships by equations. For example, if total cost t is proportional to the number of n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed ast = pn.

Ability to recognize that multiplicative relationships are proportional

2d: Explain what a point (x, y) on the graph of a proportional relationship means in terms of the

Ability to identify that a proportional relationship intersects (0,0)

7.RP.2d: (0, 0).When the value of x is 0, the value of y is 0, or (0, 0). For example:


DRAFT UNIT PLAN7.RP.1-3: Analyze Proportional Relationships and Use Them to Solve Real-World and Mathematical Problemssituation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.

Ability to determine other points using (1, r)

7.RP.2d: determine other points using (1, r), where r is the unit rate: In the ordered pair (1, r), the x-value “1” represents one standard of measurement. The y-value “r” represents a given specific quantity.

.

7.RP.3: Use proportional relationships to solve multistep ratio and percent problems. Examples: simple

Ability to build on prior experience with equivalent fractions to solve multi-step

7.RP.3: cross-product algorithm is a strategy for determining a missing value in a proportion. The proportion can be set up using either as a within ratios proportion or as a between ratios proportion, as shown in the diagrams below.


When zero electricity is being generated, zero pounds of carbon are emitted.

When zero ounces of oregano are weighed, the cost is zero dollars.

In the table “A Moving Automobile,” the x-value is time (hours). The y-value “r” is distance traveled (miles). The unit rate is r

miles per hour, or r miles1h our .

According to the graph, unit rate is (1, 50)

or 50 miles1h our . To determine other points,

multiply by 22 ,

33 ,

66 , and identify (2, 100)

or 100 miles2 hours ; (3, 150) or

130 miles3h ours ; and (6,


interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

problems with ratio and percent (see 6.RP.3c)

Ability to relate “between” ratios and “within” ratios to the cross-product and factor of change algorithms

7.RP.3: factor of change algorithm is a strategy for determining a missing value in a proportion. A unit-rate approach is used, whereby the factor of change (rate of a single unit) is established first from the given values in one ratio. Then, the missing value from the other ratio is computed by multiplying the known value in the other ratio by the factor of change.

Example using the within ratios format: Three candies cost a total of $2.40. At that same price, how much would 10 candies cost? The



within ratios proportion is: $ 2.40

3 candies = x

10 candies. To determine the factor

of change (unit cost) for one candy, divide $2.40 by 3.

The factor of change is 1

0.8 (or $0.80 for one candy), meaning that as the

number of candies increases by one, the total cost increases by $0.80. So, 10 candies ($0.80 x 10) would cost $8.00.

Evidence of Student Learning: The Partnership for Assessment of Readiness for College and Careers (PARCC) has awarded the Dana Center a grant to develop the information for this component. This information will be provided at a later date. The Dana Center, located at the University of Texas in Austin, encourages high academic standards in mathematics by working in partnership with local, state, and national education entities. Educators at the Center collaborate with their partners to help school systems nurture students' intellectual passions. The Center advocates for every student leaving school prepared for success in postsecondary education and in the contemporary workplace.

Fluency Expectations: This section highlights individual standards that set expectations for fluency, or that otherwise represent culminating masteries. These standards highlight the need to provide sufficient supports and opportunities for practice to help students meet these expectations. Fluency is not meant to come at the expense of understanding, but is an outcome of a progression of learning and sufficient thoughtful practice. It is important to provide the conceptual building blocks that develop understanding in tandem with skill along the way to fluency; the roots of this conceptual understanding often extend one or more grades earlier in the standards than the grade when fluency is finally expected.

Students solve multistep problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically (7.EE.3). This work is the culmination of many progressions of learning in arithmetic, problem solving, and mathematical practices.

Common Misconceptions: This list includes general misunderstandings and issues that frequently hinder student mastery of concepts in this unit.

Students may:Draft Maryland Common Core State Curriculum Lesson Plan for Grade 7 Mathematics December 2011 of 38


Confuse the significance of the numerator compared to the denominator.

Believe that a denominator with a greater digit automatically has a greater value than a fraction with a lesser denominator, e.g., 18 >

13 .

Rely on one configuration for setting up proportions without realizing that other configurations may also be correct (see discussion of within ratios and between ratios on page 2).

Have difficulty calculating unit rate, recognizing unit rate when it is graphed on a coordinate plane, and realizing that unit rate is also the slope of a line.

Misinterpret or not have mastery of the precise meanings and appropriate use of ratio and proportion vocabulary. Miscomprehend the difference between additive reasoning versus multiplicative reasoning.

Interdisciplinary Connections: Interdisciplinary connections fall into a number of related categories: Literacy standards within the Maryland Common Core State Curriculum Science, Technology, Engineering, and Mathematics standards Instructional connections to mathematics that will be established by local school systems, and will reflect their specific grade-level

coursework in other content areas, such as English language arts, reading, science, social studies, world languages, physical education, and fine arts, among others.

Model Lesson Plans:

MSDE Mathematics Lesson Planning Organizer

Background Information

Content/Grade Level 7 Ratios and Proportional Relationships

Unit Analyze proportional relationships and use them to solve real-world and mathematical problems

Essential Questions/Enduring Understandings Addressed in the Lesson

Essential Question: What types of problem solving situations require the use of proportional reasoning? How are comparisons used in proportional reasoning? How is proportional reasoning of geometric figures used to solve problems? What kinds of questions can be answered using proportional reasoning?

Enduring Understandings:



Recognize that a given proportional situation can be represented in more than one way Distinguish between additive and multiplicative situations Recognize two equal ratios represent a proportion Recognize and represent the connection between equivalent ratios, values in a table, and graphed

ordered pairs Proportional reasoning involves comparisons of the relationships among ratios

Standards Addressed in This Lesson 7.RP.2 Recognize and represent proportional relationships between quantitiesa. Decide whether two quantities are in a proportional relationship.b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams and

verbal descriptions of proportional relationships.c. Represent proportional relationships by equations.d. Explain what a point (x,y) on the graph of a proportional relationship means in terms of the

situation, with special attention to the points (0,0) and (1,r) where r is the unit rate.

Lesson Topic Proportional relationships that lead to the idea of similarity.

Relevance/Connections 7.NS.3 Solve real-world and mathematical problems involving the four operations with rational numbers. Note: Computations with rational numbers extend the rules for manipulating fractions to complex Fractions7.G.1 Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.7.SP.6 Approximate the probability of a chance event by collecting data on the chance process that

produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability.

7.SP.8 Find the probabilities of compound events using organized lists, tables, tree diagrams and simulation.

7.RP.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.

7.RP.3 Use proportional relationships to solve multistep ratio and percent problems.



Student Outcomes Students will: Explore multiple representations to recognize proportional relationships Represent proportional relationships to solve multi-step ratio problems

Prior Knowledge Needed to Support This Learning

Understand the concept of a ratio and use precise language and symbols to describe a ratio relationshipUnderstand unit rate and use ratio and rate reasoning to solve mathematical problems Experience with plotting points on a coordinate plane

Method for determining student readiness for the lesson

Ask students to answer the following ratio and unit rate questions.

1. There are 300 people at a school play; 90 of these people are adults and the rest are students. What is the ratio of adults to students? Write your answer in three different ways.

2. Last week Bill earned $210 for a 30-hour week. What was his hourly pay rate?What would he have earned for a 40-hour week?

Learning Experience

Component DetailsWhich Standards of Mathematical Practice does this component address?

Motivation www.bbc.co.uk/skillswise/topic/ratio-and-proportion

Students will watch an approximate 1-minute video on three different professions that use ratios. (Computer Tech company, hair salon and construction company) If this video is not available, show another one or show pictures and talk about the profession and how it uses ratios.

Warm Up/Drill Students will write ratios from pictures.

Example:

Make sense of problems and persevere in solving them

Construct viable arguments and critique the reasoning of others

Attend to precision


http://www.bbc.co.uk/skillswise/topic/ratio-and-proportion


Learning Experience


Teacher presents a photo with an image that allows students to create a variety of different ratios. “What comparisons do you see that can be expressed as ratios?

Look for and express regularity in repeated reasoning

Activity 1

UDL Components

Multiple Means of Representation

Multiple Means for Action and Expression

Multiple Means for Engagement

Key Questions

Formative Assessment

Students are presented with the following problem:

A vase holds red and white roses only. For every three red roses, there are two white roses. How many flowers might be in the vase? Students may work independently, pairs, and or small

groups to determine at least 3 different arrangements of red and white roses in a vase. Teacher may provide physical models to represent the arrangements

Teacher will compile results in a table. Results will be ordered from least to greatest in a table. See Attachment #1 Table

Students will use the original ratio of 3 red to 2 white roses to determine if other arrangements (Examples: 36:24; 12:5; 15:8 or 2:3) are equivalent by using multiplicative reasoning. Teacher and students will begin to use the term “proportion” to describe the relationship of these ratios.

Students will use a coordinate plane to graph at least 4 ordered pairs from the table. The following should be included in student’s observation and class discussion on graphs:o What patterns do we notice? Highlight the fact that:

Make sense of problems and preserve in solving problems

Reason abstractly and quantitatively Construct viable arguments and critique the

reasoning of others Model with mathematics Attend to precision Look for and express regularity in repeated

reasoning



Learning Experience


(1) points will lie on a straight line, (2) there is a 3:2 ratio (rise over run) between points, (3) which the line will continue through the origin.

o What is the meaning of (0, 0) in the context of the problem?

o How do the ratios that were not proportional relate to our graph?

After class observations and discussion, students will have:o drawn a line through the ordered pairs o described the rate of change from point to point on

the line o used points not on the line to determine if this is a

possible arrangement of red and white roses o analyzed the graph o described the graph as representing a proportional

relationship as intersecting the origin (0,0).


See Attachment #2 Formative Assessment

Three vases are described, two that are proportional to a given ratio (ex. 3:4) and one that is not. Students will need to identify the arrangement that is not proportional to the given ratio and justify their choice.

Warm Up/Drill Use Attachment #3 to identify figures that are proportionally Look for and make use of structure



Learning Experience


larger or smaller than the given figure. Students are asked to find figures that are reduced (smaller) or enlarged (larger). This should lead to a discussion where the teacher introduces the idea of similar figures.


Activity 2

UDL Components

Multiple Means of Representation

Multiple Means for Action and Expression

Multiple Means for Engagement

Key Questions


Summary

How can we use a proportion to prove figures are similar? Think –Pair–Share

Through class discussion and the activity below students will discover that you can justify that figures are similar using “between ratios” and “within ratios” (see vocabulary).

Students are given 14 rectangles drawn on centimeter paper and ask to sort into three “groups” o Group A 1• 1, 3•3, 5•5, 7•7, 10•10 o Group B 4• 1, 8•2, 12•3, 16•4 o Group C 2•1, 4•2, 6•3, 8•4, 10•5

For each group A, B, and C, students need to record in a graphing calculator the length and width of each rectangle, using the L1 function for “length” and L2 for “width”.


Reason abstractly and quantitatively Construct viable arguments and critique the

reasoning of others Model with mathematics Attend to precision Look for and express regularity in repeated

reasoning



Learning Experience


Example using Group B data:

L1 L2

0 0

4 1

8 2

12 3

16 4

Students should divide the “length” data by the “width” data using the calculator’s ability to divide lists by one another (L1/L2 and L2/L1).

Students should make observations of what numeric patterns they recognize in their tables. Discuss what the pattern from each row of data in the table tells about the proportional relationship (ratio, unit rate) between the lengths to widths (or widths to lengths) of the rectangles in each of the three groups.

Students discuss leading questions:o What is the ratio (unit rate, constant of

proportionality) for the comparative dimensions of



Learning Experience


rectangles in each group? When the length (L1) is 1?o How can we determine additional rectangles to fit in

each group?

Using data from the Group B rectangles, students graph the ordered pairs on a coordinate plane. Lead the students in the discussion of the important attributes of the line (ex. unit rate or constant of proportionality, line goes through the origin, goes in positive direction, etc.)

Draw a right triangle between each pair of two consecutive points on the line to represent the constant of proportionality (unit rate) by analyzing rise (y) over run (x).

What do you notice about the proportional relationship (rise compared to the run) of each rectangle? Teacher should ask the following questions:o For each 1 unit we move to the right on the x-axis

(run), how many units do we move up on the y-axis

(rise)? (14 )

o When we move 4 units to the right, how many units

do we move up? (4• 14 )






Learning Experience





o When we move x units to the right, how many units

do we move up? (x• 14 )

Starting from (0, 0), to get to a point (x, y) on the graph,

go x units to the right and go up x • 14 units. Therefore:

x• 14 = y OR y =

14

x

Students can complete this exercise for all the rectangle groups and write the equations to represent the line created by the constant of proportionality (unit rate) for the rectangles in Group A and Group C.

Students explain how they could determine a missing dimension of a rectangle when one dimension is given in order to maintain similarity with the other figures in the group. Students can represent their method by writing an equation/proportion.

Give students this rectangle



Learning Experience


Ask students to name the length and the width of 3 other rectangles that are similar to this one and graph the dimensions of each. Ask students to discuss the important attributes of the line (ex. unit rate or constant of proportionality, line goes through the origin, goes in positive direction, etc.)

Discuss the difference between congruent and similar figures. Can they ever be both?

Attachment 4 Similar Figures Students will examine a pair of figures to determine if the figures are congruent, similar or neither. Students formulate the definitions for similar and congruent figures. Share outcomes with the whole class.

(Note to teacher: Depending upon your class, you may want to add more to this sheet or give them models to extend this activity.)

Formative AssessmentStudents will choose 1 rectangular “group” and create another



Learning Experience


rectangle that would belong to that group and justify their answer using a proportion, graph and/or equation in their justification.

Supporting Information

Interventions/Enrichments

Special Education/Struggling Learners

ELL Gifted and Talented

Activity 1: Vase

Provide manipulatives such as colored chips for students to build their flower arrangements. Differentiate the given ratio of “3:2” by using a 1:2 ratio for struggling students or familiar numbers to ELL students or a 5:7 ratio for students with higher mathematical abilities.

Activity 2:Students may be assigned two out of the three rectangular “groups” to study.Activity may be presented as a Jigsaw, where small groups are assigned one rectangular group to complete the activity then come back as a whole class to present findings and make connections between rectangular “groups”.

Gifted and Talented students can place the data from L1 and L2 into a scatter plot on a graphing calculator. Through observation of the data, these students may be ready to determine an equation that represents each proportional relationship, and then graph it on their calculators to see how the scatter plot compares to the line of the equation. Also, they can check the relationship by using the “TRACE” function to compare the x-values



with their y-values and see if they match the values from their original lists.

Materials Manipulatives, chips, beans, connecting cubes, color tiles, etc.Graph paper, rulerCopies of Attachments 1, 2 and 3Diagrams of rectangles in groups A, B and C (teacher will create.)Graphing calculators

Technology Document cameraComputer and projection unit for Motivation video

Resources

(must be available to all stakeholders)

www.bbc.co.uk/skillswise/topic/ratio-and-proportion

Litwiller, B., Bright, G. (2002). Making Sense of Fractions, Ratios, and Proportions, Reston, VA: National Council of Teachers of Mathematics


http://www.bbc.co.uk/skillswise/topic/ratio-and-proportion

DRAFT UNIT PLAN7.RP.1-3: Analyze Proportional Relationships and Use Them to Solve Real-World and

Mathematical ProblemsAttachment #1 Table

A vase holds red and white roses only. For every three red roses, there are two white roses. How many flowers might be in the vase?

Step A: Try to come up with three different arrangements of flowers for the vase. 1. 2. 3.

Step B: Record class data below. Step C: Order the data from least to greatest.

Red White

Step D: Graph the data on your graph paper.


Red White


Mathematical Problems

Attachment #2 Formative Assessment

Arrangement 1 Arrangement 2 Arrangement 3

Sarah has 9 red and 12 white flowers in her vase.

John has 6 red and 8 white flowers in his vase.

May has 12 red and 8 white flowers in her vase.

Which employee(s) did not follow the directions? ________________________________________

Justify your answer.


Your boss at the flower shop has decided that all of the new flower arrangements need to have a ratio of 3 red flowers to 4 white flowers. The three arrangements were made by three different employees. Did the employees follow their boss’ directions?

http://www.google.com/imgres?q=vase+with+red+and+white+flowers&hl=en&sa=X&biw=1249&bih=545&tbm=isch&prmd=imvns&tbnid=f1gCP4rEmBZutM:&imgrefurl=http://www.punjabflowers.com/flowervases.html&docid=02an8YGIoNdUyM&imgurl=http://www.punjabflowers.com/product_images/vase9.jpg&w=275&h=275&ei=YnzGTuTNFILY0QGz59Aj&zoom=1


Mathematical ProblemsAttachment #3 Similarity

Circle any figure on the right that is a reduction (smaller) or an enlargement (larger) of the figure on the left.

1.

2.



Mathematical Problems

3.


DRAFT UNIT PLAN 7.RP.1-3: Analyze Proportional Relationships and Use Them to Solve Real-World and Mathematical Problems

Attachment #4 Similar Figures

Directions: Look at the figures in the first column. Then put a check mark in the correct column showing whether you think the figures are similar, congruent or neither. In the last column, explain why you think your choice is correct.

Draft Maryland Common Core State Curriculum Unit for Grade 7 Mathematics December 2011 of 38

FIGURES similar congruent neither Explain Why


Lesson Seeds: The lesson seeds have been written particularly for the unit, with specific standards in mind. The suggested activities are not intended to be prescriptive, exhaustive, or sequential. Rather, they simply demonstrate how specific content can be used to help students learn the skills described in the standards. They are designed to generate evidence of student understanding and give teachers ideas for developing their own activities.

Seed #1 - Domain: Ratios and Proportional RelationshipsStandard: 7.RP.2. Recognize and represent proportional relationships between quantities

Purpose/Big Idea: Investigate direct variation to decide whether two quantities are in a proportional relationship by testing for equivalent ratios in a table or graphing on a coordinate plane; identify the constant of proportionality (unit rate) in tables and graphs of proportional relationships.

Materials: (1) graphing calculator – preferably TI-83/84 Plus Family & TI-73 Explorer

Activity: Using a graphing calculator, students will solve problems involving direct variation and will identify relationships among values through the use of tables, scatter plots, and graphs on a coordinate plane; and students will apply direct variation, proportion, and percent to authentic scenarios.

Resource: Dase, P.H. (2007). Texas Instrument Graphing Calculator Strategies: Algebra (pp. 49-56).Huntington Beach, CA: Shell Education

Guiding Questions: What does “direct variation” mean?How does the concept of “direct variation” relate to or exemplify a ratio or a proportional relationship? What is the constant of proportionality (unit rate) for a given data set?When given data set is graphed on a scatter plot, what is the relationship between the resulting points?How does this relationship relate to a ratio? To a proportion? To a unit rate?

Seed #2 - Domain: Ratios and Proportional RelationshipsStandard: 7.RP.2. Recognize and represent proportional relationships between quantities

Purpose/Big Idea: Decide whether two quantities are in a proportional relationship by testing for equivalent ratios in a table or graphing on a coordinate plane; identify the constant of proportionality (unit rate) in tables and graphs of proportional relationships.

Materials: graphing calculator

Students may use a content web site and/or interactive white board to create tables and graphs of proportional or non-proportional relationships. Graphing proportional relationships represented in a table helps students recognize that the graph is a line through the origin (0,0) with a constant of proportionality equal to the slope of the line.

Activity: A student is making trail mix. Create a graph to determine if the quantities of nuts and fruit are proportional for each serving size listed in the table.



Resource: http://www.azed.gov/standards-practices/mathematics-standards/

Guiding Questions:If the quantities are proportional, what is the constant of proportionality or unit rate that defines the relationship? Explain how you determined the constant of proportionality and how it relates to both the table and graph.

Sample Assessment Items: The items included in this component will be aligned to the standards in the unit and will include: Items purchased from vendors PARCC prototype items PARCC public release items Maryland public release items

Interventions/Enrichments/PD: (Standard-specific modules that focus on student interventions/enrichments and on professional development for teachers will be included later, as available from the vendor(s) who will produce the modules.)

Vocabulary/Terminology/Concepts: This section of the Unit Plan is divided into two pints. Part I contains vocabulary and terminology from standards that comprise the cluster which is the focus of this unit plan. Part II contains vocabulary and terminology from standards outside of the focus cluster. These “outside standards” provide important instructional connections to the focus cluster.


Serving Size 1 2 3 4 Cups of Nuts (x) 1 2 3 4Cups of Fruit (y) 2 4 6 8

http://www.azed.gov/standards-practices/mathematics-standards/

AB

ab

Within Ratios Within Ratios


Part I – Focus Cluster: Ratios and Proportional Reasoning

unit rate: Given a specific quantity in dollars, inches, centimeters, pounds, hours, etc., the unit rate is the amount for one standard of measurement (i.e., one dollar, one inch, one centimeter, one pound, one hour, etc.) relative to the given quantity.

ratio: A ratio is the quotient of two numbers (or quantities); the relative sizes of two numbers (or quantities). The three ways of writing a

ratio include: 9 to 10, and 9:10, and 9

10 .

complex fraction: A complex fraction has a fraction for the numerator or denominator or both.

proportional relationship: A proportional relationship is a collection of pairs of numbers that are in equivalent ratios (i.e., 30 miles2 hours =

60 miles4 h ours ). A proportional relationship also can be described by an equation of the form y = kx, where k is a positive constant (often

called a “constant of proportionality”).

within ratios: A ratio of two measures in the same setting is within ratios.


Within ratios: AB =

ab or

BA =

ba

Example: If a person walks 4 miles in 2 hours, how many miles will she walk in 3 hours? Let m = miles

Set up of the proportion using “within ratios” format:

42 =

m3 or

24 =

3m

AB

ab

Between Ratios

Between Ratios


between ratios: A ratio of two corresponding measures in different situations is considered between ratios.

additive reasoning versus multiplicative reasoning: Proportional situations are based on multiplicative relationships. Equal ratios result from multiplication or division, not addition or subtraction.

Consider any proportion, for example 17 =

535 . What operation was used to convert

17 to

535?

Was additive reasoning used: 17 +

55=

4035 ? Or was multiplicative reasoning used:

17 •

55=

535?

The example below shows an application of multiplicative reasoning:

Last month, two bean plants were measured at 9 inches tall and 15 inches tall. Today they are 12 inches tall and 18 inches tall, respectively. Which bean plant grew more during the month, the 9-inch bean plant or the 15-inch bean plant?


Between ratios: Aa =

Bb or

aA =

bBExample: If a person walks 4 miles in 2 hours, how

many miles will she walk in 3 hours? Let m = miles

Set up of the proportion using “between ratios” format:

23 =

4m or

32 =

m4


Using additive reasoning, we know that each plant added 3 inches in a month. To determine which plant grew more, however, we use multiplicative reasoning. In other words, we want to know what proportion of the original plant height is represented by 3 inches.

The 3-inch increase represents 39 or

13 of the first plant’s growth, whereas the 3-inch increase represents

315 or

15 of the second

plant’s growth. Since 39 >

315 (or

13 >

15¿, the 9-inch plant grew more.

proportion: A proportion is a statement of equality of two ratios; an equation whose members are ratios.

identify that a proportional relationship intersects (0, 0): As shown on the graphs below, when the value of x is 0, the value of y is also 0, or (0, 0). For example, (a) when zero electricity is being generated, zero pounds of carbon are emitted; (b) when zero ounces of oregano are weighed, the cost is zero dollars; (c) when a cone has a height of zero, the cone flattens into a 2-dimensional circle, and thus, has volume of zero.

a. b. c.

determine other points using (1, r), where r is the unit rate: In the ordered pair (1, r), the x-value “1” represents one standard of measurement. The y-value “r” represents a given specific quantity.


For example in the table “A Moving Automobile,” the x-value is time (hours). The y-value “r” is distance traveled (miles).

The unit rate is r miles per hour, or r miles1hour . According to the

graph, unit rate is (1, 50) or 50 miles1 hour . To determine other

points, multiply by 22 ,

33 ,

66 , and identify (2, 100) or


percent error: We often assume that each measurement we make in mathematics or science is true and accurate. However, sources of error often prevent us from being as accurate as we would like. Percent error calculations are used to determine how close to the true values our experimental values actually are. The value that we derive from measuring is called the experimental, or observed, value. A true, or theoretical, value can be found in reference tables.

The percent error can be determined when the theoretical value is compared to the experimental value according to the equation below:

percent error = experimental value−t heoreticalvalue

t h eoretical value x 100%

Example: A student measured the volume of a 2.50 liter container to be 2.38 liters. What is the percent error in the student's measurement?

percent error = 2.50liters−2.38 liters

2.50 liters x 100%

= 0.12liters2.50liters x 100%

= 0.048 x 100% = 4.8%

cross-product algorithm: This algorithm is a strategy for determining a missing value in a proportion. The proportion can be set up using either as a within ratios proportion or as a between ratios proportion, as shown in the diagrams below.



factor of change algorithm: This algorithm is a strategy for determining a missing value in a proportion. A unit-rate approach is used, whereby the factor of change (rate of a single unit) is established first from the given values in one ratio. Then, the missing value from the other ratio is computed by multiplying the known value in the other ratio by the factor of change.

Example using the within ratios format: Three candies cost a total of $2.40. At that same price, how much would 10 candies cost?

The within ratios proportion is: $ 2.40

3 candies = x10 candies . To determine the factor of change (unit cost) for one candy, divide $2.40

by 3. The factor of change is 1

0.8 (or $0.80 for one candy), meaning that as the number of candies increases by one, the total cost

increases by $0.80. So, 10 candies ($0.80 x 10) would cost $8.00.

Part II – Instructional Connections outside the Focus Cluster

rational numbers: Numbers that can be expressed as an integer, as a quotient of integers (such as 12 , 4

3 , 7), or as a decimal where the

decimal part is either finite or repeats infinitely (such as 2.75 and 33.3333…) are considered rational numbers.

probability: Probability is used to describe the likeliness that an event will happen.

chance event: An event in probability is one of a collection of possible outcomes. When a coin is tossed, “heads” and “tails” are chance events that can possibly occur. When a baby is born, “male” and “female” are the chance events. When tossing a number cube, 1, 2, 3, 4, 5, and 6 are the chance events.



relative frequency: When a collection of data is separated into several categories, the number of items in a given category is the absolute frequency. The absolute frequency divided by the total number of items is the relative frequency. Out of 50 middle school students,

18 are sixth graders; 18 is the absolute frequency. The relative frequency is 18 six h grade stude nts

50 total students or 0.36

outcome: A possible end result of a probability experiment is referred to as an outcome.

event: An event is a collection of possible outcomes.

simple event: An event is a collection of possible outcomes. A simple event is a single outcome of an experiment. For example, tossing two number cubes and getting a 7 is an event. Getting a 7 specifically with a 2 and a 5 is considered a simple event.

compound event: An event is a collection of possible outcomes. An event that consists of two or more events is a compound event. The probability of a compound event can be determined by multiplying the probability of one event by the probability of a second event. Some compound events (independent events) do not affect each other's outcomes, such as rolling a number cube and tossing a coin. Other compound events do affect each other's outputs (dependent events). For example, if you take two cards from a deck of playing cards, the likelihood of second card having a certain quality is altered by the fact that the first card has already been removed from the deck.

sample space: Sample space is the collection of all possible outcomes in a probability experiment.

simulation: Simulation is a technique used for answering real-world questions or making decision in complex situations where an element of chance is involved.

Resources: This section contains links to materials that are intended to support content instruction in this unit.


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